# Message #3394

From: apturner@mit.edu

Subject: Re: Cube puzzles and math

Date: Mon, 27 Jun 2016 13:07:21 -0700

Dear Sid,

The notation for the operation doesn’t really matter. The issue is that without another binary operation, the object you’re talking about cannot be a ring. A ring is a set together with two binary operations, one of which forms an abelian group with the set, and the other of which is associative and distributes over the group operation, but may or may not be commutative (and may or may not have an identity element, depending on how you define a ring). As far as I am aware, there is no way to turn the Rubik’s Cube group into a ring in any way that relates to the physical reality of the puzzle.

As for whether the groups are the same or not, my point was simply that the group Joel refers to and the group that you are actually defining in your post are isomorphic, which means that they are really just different ways of talking about the same mathematical structure. But it’s true that sometimes it’s good to be careful and actually acknowledge that two groups are not truly the exact same group, but rather isomorphic, so I will concede that point.

Cheers,

Andrew